Embed Size (px)
Citation preview
t0
RADC-TR-77-243 __
'IN-HOUSE REPORT
JULY 1977
Detection of Targets in Noise and Clutter
RONALD L FANTE
""AppAro d for public release; distribution unlimie..
ROME AIR DEVELOPMENT CENTER__ AIR FORCE SYSTEMS COMMANDL.. GRIFFISS AIR FORCE BASE, NEW YORK 13441
This report has been reviewed by the RAflC InformationOffice (01) and is releasable to the National TechnicalInformation Service (NTIS). At NWIS it will. be r'eleasable tothe general public, including foreign nations.
APPROVED:
Chief, Microwave Detection Techniques Br.Electromagnetic Sciences Division
APPROVED;
ALL- C. SCHELLActing ChiefElectromagnetic Sciences Division
FOR TIP COMiANDER:e
Plans Office
UJrlasgifiedSECURITY CL.ASS.FICA~tON OF THIC 04, fUA., no IEnI..d) _________________
REAL' INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFC1-RC COWIPLEriNG FORM
- - -I/ ~11 OVY ACCESSION NO. 3- PECI-ri, TS CAT &LCOG"SE
_ETECTION OF TARGETS IN NOISE
FRonald L./Fante
9. PERFORMING ORGANIZATION NAME AND ADDRESS/7 -A ELEMENT. PROJECT. T
Depuy fo Elctroic echrolo RAD/ETE) 4 b N4JRKUNIT NUMOERS/
Massachusetts 01731 ESe
Deputy for Electronic Technology (RADC/ETEP Jui 7 2Hanscom AFB
14. MONITORING AGENCY NAME 4 AOISRESSrfd(I -1 df lI- -- 1- i i S.) . SECUIYCLAE SSI(of t his~pGAO
Unclassified
SCHEDULE
16. OISTRIaUTIOIJ STATEMENT (of this R~port)
Approved for public release; distribution unlimited.
f ~17. DISTAISSTICH STATEMENT (oilthom.1.,act .. ,te,sdi Bl Rock 20. It E4ffto-I It", Ron,-,l
10. SUPPLEMENTARY NOTES Uf .- 4
19. KEY WORDS (I-.. ld .... dI-ly yb- - -
ClutterDetectionRadar
20 AOSSRACT (C-~II... I- -* . 1.. d. .'it .- d Id.olIfy by block .- ~b.,)VWe have calculated the probability of detecting eitl-er a Swerling-l or
Swerling-2 target immersed in both Rayleigh noise and log-normally distributelclutter. Re ults are presented which indicate the effect on detection of thenumber of WlJes integrated, signal -to -noise ratio and the noise-to -clutterj ~ ~~~~~ratio.~ EIINO OA SOEC SCRT LSIIAINO hSPG . 'r f,,d
DD I OR, 1473 EDTO FINV6 SONOEEUnclassifiedLEUIY.ASFCTU FTISPG 1tf o.Et~d
SECJRITý CLASSIFICATION OF THIS PAOR(WhSm Date Entre.d)
t
$1ECuRImTY CLASSlIFICATION~ OF THIS1 P•AG(W'hon Dat Ifo •noed)
I .. .. " ' | . . . .- .. . .. . .. " "' n. . " . . .' " . . -, • ' .-.... ."
Preface
The author is grateful to Dr. Peter Wintersteiner of ARCON Corp. who pro-
gramined the integrals in Equations (35)-(37).
ii
or "• '" F;=r
I -j
t. .., . . ...........
II N:IT
IoY~b:./lu , ' ... SItCIAL
.. I..
3
1. INTRODUCTION 9
2. THEORETICAL ANALYSIS 102. 1 Target and Clutter Constant From Pulse-to-Pulse but
Varying From Scan-to-Scan (Sweriing 1) 102. 2 Clutter Varying From Scan-to-Scan but Target Varying
From Pulse-to-Pulse (Swerling-2 Target) 15
3. PROBABILITY OF FALSE ALARM AND DETECTION 18
4. NUMERICAL RESULTS 20
4.1 False-Alarm Probability 204. 2 Probability of Detection for Swerling-2 Targets 234. 3 Probability of Detection for Swerling-1 Targets 28
5. DISCUSSION 32
REFEPENCES 33
APPENDIX A. Probability Density for a Constant Target 35
APPENDIX B. Derivation of Equation (38) 37
APPENDIX C. Derivation of Equation (40) 39
I°
ALAW
5-
V - -_'-
U•-.
Illustrations
1. False-Alarm Probability for N = 1, a z 0.7 21
2. False-Alarm Probability for N = 3, a z 0. 7 21
3. False-Alarm Probability for N = 6, a = 0.7 21
4. False-Alarm Probability for N 6 10, a = 0.7 21
5. Effect of a on Pf for 3.16, N = 1 22
6. Effect of a on P for 3.16, N = 10 227. Probability of Detection (Swerling 2) for Pf36 10-4,N1002
7. Pobailit ofDetetio (Swerlng ) fo P 0 0316,a = 0.7 23
8. Probability of Detection (Swerling 2) for P 10-4, 1.0,a =0.7 f 23
9. Probability of Detection (Swerling 2) for Pf = 10-4, 33.16,a = 0.7 23
10. Probability of Detection (Swerling 2) for P1 = 10-14 0,a 0.7 23
11. Probability of Detection (Swerling 2) for P = 104 100,a =0.7 25
12. Probability of Detection (Swerling 2) for P = 10-, 0. 316,a =0. 7 f2512.
31. Probability of Detection (Swerling 2) for P 10' 1,0=-0.7 25
14. Probability of Detection (Swerling 2) for Pf = 10 6 , 3. 16,a =0.7 25
15. Probability of Detection (Swerling 2) for Pf 10-6, j3 10,a = 0.7 26
16. Probability of Detection (Swerling 2) for P = 10- 1 100,a= 0.7 26
17. Effect of a on Pd? for Pf = 10, f3 = 3.16 26
18. Effect of a on Pd2 for Pf = 10 -, =3.16 26
19. Probability of Detection (Swerling 1) for Pf - 10-4, (3 0.316,a . 7 29
-420. Probability of Detection (Swerling I) for Pf = 10-, (3 3 1,
a =0.7 29
21. Probability of Detection (Swerling 1) for P = 10"-. 3.16,a 0. 7 f 29
22. Probability of Detection (Swerling 1) for P1 = 10-, 100=0.7 29
23. Probability of Detection (Swerling 1) for P -- 10-4,•- 100a = 0. 7 f30
24. Probability of Detection (Swerling 1) for Pf = 10 0. 316,0=0.7 30
25. Probability of Detection (Swerling 1) for Pf = 10 o 1,a= 0.7 30
6
Illustrations
26. Probability of Detection (Swerling 1) for P f 10 -6 3 3. 16, 30a = 0.73
27. Probability of Detection (Swerllng 1) for P f 0 - 10,0 0.7 31
-628. Probability of Detection (Swerling 1) for P 1 , 3 0 100,
a =0.7 31
Tables
1. Comparison of Pf as Calculated From Equation (38) With theExact Result 22
2. Valve of Signal-to-Noise Ratio a Required for Pd2 - 0. 9, 0. 99 and0. 999 When Pf = 10-6 and o = 0.7 27
3. Comparison of Pd2 Calculated Prom Equation (40) With the ExactResult foro a 0.7 and Pr 0-6 28
,I
7
LijI
)1
iI
Detection of Targets inNoise and Clutter
1. INTRODUCTION
The probability density function (after detection and video integration) for a1fluctuating target in gaussian noise has been known for some years now. Similarly,
results have also been obtained for the case of a fluctuating target immersed in log-
normally distributed clutter.2,3 However, it does not appear that any results have
been obtained for the case of a fluctuating target immersed in both noise and clut t er.
Therefore, in this paper we will calculate the (post square-law detection) probabil-
ity density function for the sum of N pulses, after video integration, for the case of
a target immersed simultaneously in gaussian noise and log-normally distributed
clutter. We shall consider principally the case when the clutter power is of the
same order of magnitude as the noise power, because when the clutter power is
very much larger than the noise power the results in2 are appropriate, whereas
when the noise power greatlý exceeds the clutter power the results in 1 are adequate.
(Received for publication 21 July 1977)
1. Marcum, J. . and Swerling, P. (1960) Studies of target detection by pulsedradar, IRE Trans. on Inform Theory IT-6:2.
2. Trunk, G., and George, S. %1970) Detection of targets in non-gaussian seaclutter, IEEE Trans. on Aerospace and Electronic Systems AES-6:620.
3. Schleher, D. (1975) Radar detection in log-normal clutter, IEEE InternationalRadar Conference, IEEE Press (Publication No. 75 CH0938-1 AES),New York.
9
" •6it
We shall present detailed numerical results for the probability of a false
alarm and the probability of detection for both Swerling-l and Swerling-2 targets.
2. TIIHIHOETICAL, ANALYSIS
2.1 T'•rgvt and Clutter .:on.tant 1:41m Ilul.e-to-Pulbe but A uryingFroin SCtn-to-Sv.,n (Swerlinjg 1)
Let us suppose we have a target immersed in gaussian noise and log-normally
disiributed clutter, and this target is to be detected using an N-.pulse radar burst.
If we assume thac both the target and the clutter are constant from pulse-to-pulse
but vary from scan-to-scan, then the single pulse return R•e ican be written as
Re' p + rei (1)
where p represents the target-plus -clutter and r exp(ihb) is a complex phasor rep-
resenting the noise, which is assumed to vary from pulse-to-pulse, and hase the
probability density (for r 7 0)
For mathematical convenience, we assume that the signal is to he detected
utsing a square-law device. W,,e are therefore interested in the probability distr-ibu-tionl Qf the square, R2 , of the envelope of the received signal. If we define
p2
X 2 2
n n
and temporarily treat p as though it were a sonstantt because it d'es not vary frompulse-to-pulse, it is readily shown4 that the conditional probability pr y p ix) i
4. Beckmann, P. (1967) Probabilit in CommunicationEngineerina , we areourt,
Brace and World, New York
10
[i/2%i~~~ P:-, .L
p(ylx) = ep(-x y) 10 (2.Iy@ for y _ 0
: 0 for y < , (3)
where I. (...) is the modiFied Besse] function. The characteristic function C (slx)
associated with p(ylx) is
C C(aslx) J dye -'y p(yx)
fx dy e*(1)y 10(2/ ITO (4)
04x
(S 4 1 exp x +
For N-pulse detection, we detect the sum of N differen: values Y of the sig-lal
"plus noise variables. That is,
S~~Y1 ' Y3 +2 ' Y- " + YN (1.=
If the noise samples ate statistically irdepLndent, then the characteristic functionof the sum of the N samples is the product oj their individual characteristic func-
tions. Therefore, the characteristic function CN((6Ix) associated with i is
(sX s+1 x x1(6)I.Now because the target and clutter fluctuate from scan-to-scan, we must
therefore average (6) over the variable x p /2in That is
C (S) dx p(x) Ce(-gIx )
N f_0 x p(x) 1t (7)+
(S+ N
• •1"
where p(x) is the probability density of the random variable x. We shall calculate
this later. Once C N(c) is known, the probability density of Y y1 + Y2 " " N is
(Y) f ds CN(s) esY
0 0
f dx p(x) e-Nx GN(x. Y) (8)
where
GN(X.Y) j --1 (N- 10 (2,,/ ) -IN_1 (2 TTx (9)
The result in Eq. (9) follows from the (N - 1) fold differentiation of both sides of
the known result
ds exp (sY + NX -
2si f+ (s • = 10 (2/'R'x-Y) (10)
*1
In order to evaluate Eq. (8) for p(y), we must first obtain p(x). We recall
that p is the sum of the target and clutter and car, be written as
p = j exp (ix) + Y exp (liv) (11)
where p exp (iX) is a phasor representing the target return and ? exp (iq,) is a
phasor representing the clutter signal. For a Rayleigh target, we know that
Ptlu. X) '--P- exp for IŽ _0 (12)2 7tot a
:0 for U <c0c
12
and the log-normal clutter is governed by the probability density
r 2
p (n. 0 (2w)"3/2 (001" exp n-2 for Yt0 (13)
=0 for r)<O
The quantity rl in (13) is the median value of Y1. Also, the standard deviation a"0 2, 3has been estirnated to be 0.7 for low-se, -state clutter, 1. 1 for ground clutter,
and as high as 1. 6 for high-sea-state clutter.
The probability density for the envelope of a Rayleigh plus log-normal phasora 4
is given in Chapter 4 of Beckmann and can be written as
13 c()p exp dJLr jdF(7P)I40~j for p 2-0 (14)
=0 for p <O0
where [ I n - 22 2
If we rcall that x p /2an. it is easy to show that
a roW Pt a 2n e-bxdr F(- I 2x11/2 •f for x 0 (15)
o otj
where
b a I(n/at1
We nok use (15) in (8) and set Y - r//ot and After !3ome manipulation.
we obtain
13
P ,(Y) Y a eY (YýN-lf)12 d9 I N-1 (2NY)r -e-0l41A) M(Q) of-2for YI0
0(16)
-0 for Y<O
where
(2 r )1 2 2 8- -8 2
1SN-(18)
_no noise to clutter ratio (19)
t signal to noise ratio (20)
Equation (16) is the probability density for the sum of N pulses (following square-
law detection) received from a slowly varying target immersed in slowly varying
clutter but rapidly varying noise. This can be thought of as a Swerling-1 target
immersed in Rayleigh noise and Swerling-1 clutter.
As a check on the validity of (16), we can study the limit as the clutter power
vanishes (03 ct). In this case, (17) should reduce to the usual Swerling-1 result.
When B -ý oc it is clear that the only contribution to the integration over n in (17)
comes from q very near zero. Therefore, we can expand the integrand in a
Taylor series about q = 0. This gives
[0 2M(9) 1 Y exp In (a13 l
=00 (c2 7T) a- I. 82
14
IIi If we set M(ý) 1 in (16) we readily find using p. 187 of Bateman5 that forYa 0
N 2_-:(
where y(N, x) is the incomplete gamma function defined on p. 387 of Bateman.
Equation (21) is precisely the well-known result for a Swerling-l target in Rayleigh
noise, as is seen by comparison with the results in Section 3. 5 of Berkowitz. 6
An additional check is obtained by studying the result for N = 1. For y _ 0,
(16) becomes in this case
-y exp ( _yF 11/2 2P ((Y) 1 1 2 a(, + "- n (2 exp - in((20-) c~ o21 + ) 8G2
(22)
which is the well-known result for the sum of two Rayleigh and one log-normal
random variables. Equation (22) is also appropriate for the N-pulse case in the
limit when the noise samples are completely correlated from pulse-to-pulse, as
opposed to the case we have considered here with the noise samples statistically
independent from pulse-to-pulse.
2.2 Clutter varying From Scan-to-Scan hut Target %arying FromPulse-to-Pulse (Swerling-2 Target)
When the target varies so rapidly that its cross section changes from pulse-to-
pulse (Swerling-2 Target) but the clutter is constant from pulse-to-pulse (where
the clutter is assumed to vary from scan-to-scan) we can replace Eq. (1) by
Re - u + v e(23)
where u represents the clutter and v exp (i6) is a phasor representing the target
plus noise. For a Rayleigh target immersed in Rayleigh noise (that is, the noise
envelope is Rayleigh, although the quadrature components are gaussian), the joint
probability density of v and 6 is given by (for v L- 0)
5. Erdelyi, A., et al (1954) Tables of Integral Transform (Bateman ManuscriptProject), McGraw-Hill, New York.
6. Berkowitz, Ii. (1965) Modern Radar, Wiley, New York.
15
I.
Ptn(V, 6) 2 +02 exp 2 V 2 (24)
22AGon at) 2(n + at)
where is the noise variance and 2 is the target variance. We are againhee 2n istenievra c n t
interested in obtaining the probability density of the square of the envelope of the
total received signal. Let us define
u2Z = 2 2
2 (a n +
R2 y
yt + a
where y is the same variable defined fullowing Eq. (2) and a is defined in (20). If
we temporarily treat u as a constant because it does not vary from pulse-to-pulse
(but only from scan-to-scan), it is straightforward to show4 that
p(y'jz) = exp (-z - y') I0 [2(zy')l/2] for y' _ 0
(25)
= 0 for y' < 0
which gives a single-pulse characteristic fumction
C1 (sI z) = (s + 1) " exp ( + (26)
For N-pulse detection, we detect the sum of N different values y' In partic-
ular
" ' + y ... • . (27)
If we assume that the target plus noise variables are statistically independent from
pulse-to-pulse, the N-pulse characteristic function is the product of the single
pulse ones and we get
16
[-N Is + - -
CN(SI z) (s + 1)-N exp N-. Nz z (28)
Because the clutter varies from scan-to-scan, we must next average (28) over
the clutter probability density. This gives
Ndz pc(z) exp Nz+ I NzoO (S N / (29)CN(S f (s + 5
where pc (z) is the probability density of the square of the clutter envelope. If we
write the clutter u as the random phasor u = n exp (ig), it is evident that p. (r) k) is
given by (13). In terms of the variable z = u 2 /2(a2 + 02), this becomes
Pc(z) = [2 (I2 ]i) [. enxpz) for zŽ (30)
0 for z<O
2where h W (on+ a )/To. If we use (30) in (29). we getn t 0
S(2) exp -Nd + Nz _ z (31)CN(S) 1/2 •(s + 1 )N J s
so that
tt A~+ioo eY
p = J dse t CN(S)
A-ioo
is given by
dz,) en 2(2hz) 1 Y Y' 0
N 2 2(27r)1/2 f -- ex ) for 80
(32)
=0 for Y' < 0
17
..-. <
S...... r.,. • "u.- ¶'YW..• 4.s -.' [.'•., . t -•,. '"•, •,.
where GN(z, Y') is given by (9). Finally, if we let C = Nz we can rewrite (32) as
xy)N )/2 g- 0 I 1/2 In Ni h
PN = 2(2)1/2 a f d9UV2 1 12(Y'CY er' -
for Y' I e0 (33)
-0 for Y' < C
As a check on the validity of (33), we can consider the limiting case when the
clutter is absent (b - C). In this case, it is evident that the dominant contribution
to the C integral comes from g near zero. Therefore, we may expand the integraildin a Taylor series about C =0. If this is done, the E integral may be performed
and we find (for Y' Ž_ 0)
(N-1 -Y'
(N - 1)!
which is identical with the well-known result for the N-pulse sum for a Swerling-2
target m Rayleigh noise.
3. PROBABILITY OF FALSE ALARM AND DEFECTION
The probabiiity of a false alarm is the probability that the received signalexceeds some threshold level Yo when there is no target present. In the limit
when the target is absent, it is readily demonstrated that for Y '2 0, (16) becomes
Sy-l)/2 e-Y J Nn [ / x( ) ( r e f d I N _ ---g, - x p- ':1, ) .
04 ((34)
18
Therefore, the probability of a false alarm for an N-pulse detection system is
Pf 2(27)1120 Y e~(~ 1)/ 12
"ai
L. MI1FRIC_'AL RE'UILTS
Before presenting our results it is useful to reemphasize once again ourdefinitions for signal-to-noise and noise -to -clutter ratios, since they may differ
somewhat from those used by other authors. They are
2at
-- '-- signal-to-noise ratio,
n
2n
-- = noise to median-clutter ratio,lie
N= number of independent pulses integrated,
P f 'probability of a false alarm,
Pdl z probability of detecting a Swerling-l target in noise varying frompulse-to-pulse and clutter varying from scan-to-scan,
Pd2 = probability of detecting a Swerling-2 target in noise varying frompulse-to-pulse and clutter varying from sc.n-to-s(an.
4.1 Fah-e-Alarm Probability
By using (35), we have calculatedi the probability Pf I f a false alarm as a
function of the threshold Yo = Ro/2n , for a number of different values of thenoise-to-clutter ratio '3. These results, with the standard deviation a of the log
normal distribution set equal to 0.7, are presented in Figures 1-4. We observe
t'at the probability of a false alarm is significantly affectcd by the value of j. It
is even more strongly affected by the value of a as can be seen from the resu - in
Figures 5 and 6.
From the nature of the curves in Figures 1-6 it is clear that for large values
of Yo (that is, small false alarm probabilities), Pf is a frnction of j3Y0 /N. It is
demonstrated in Appendix B that for Y large, we may approximate (35) by
f2 12 3/2 Na0P 1- erfc [ n (231, 38)
where erfe is the compiementary error function defined by
erfc(z) 2l2 eJ 2 dt - -exp(-z 2 ) (39){0 1 / 2 fr)
[ 2 a
20
I yi10" lO-c
SN ©IN ,3
o' 07 - o-0i
13 'tO"• : )"3t310331
B.10 3 tO3 'C" 0 8-10
10toI0 10" 10 10•
YO Y0.
Figure 1. False-Alarm Probability for Figure 2. False-Alarm Probability for: N = 1 a 0.7 N =3, a =0.7
10"4 1 N 0..
N*'N6 N N10
q- 07 o-,07
o. 316
I0 z II0"0'0R. 03160100
a. 0. 0.316
CV' *~3161.0
lo-6 ~ ~ ~ Y .1 0 ..
to 1 0 YO 10"0 0 10 o YO 10 1C)
Figure 3. False-Alarm Probability for Figure 4. False-Alarm Probability forN = 6, a = 0.7 N = 10, ai 0.7
21
t °" I
100
0.11N 1010f
10" (3"' II104
a07
0~ 7~
I Q07
10 1010-5
1O i01 iO 101 102 iO5 'o, t0Y O - 't o - .•
Figure 5. Effect of o on Pf for f3 3. 16, Figure 6. Effect of a on Pf for j3 3. 16,
N= 1 N= 10
Table 1. Comparison of Pf as Calculated From Equation (38) With theExact Result
_ _PfPf (calculated from
a N 3 Y (exact) Equation (38))
S0. 7 1 1 300 2. 55 X 10- 2. 57'X I0-
0.7 1 1 500 4.20 X 10-7 4.20 X 10-7
0.7 3 I 3000 1.75 X 10-6 1.79 X 10- 6
0.7 10 1 5000 4. 10 X 10-7 4.20 X 10-7
1.1 1 I10 4000 2.0 X 10-7 2. 12 X 10-7
The final approximation in (39) is valid for z large, which is usually the case of
interest. A comparison in Table 1 of numerical results calculated from (38) with
the curves in Figures 1-3 indicates that (38) is an excellent approximation to (35)
as long as Y is large (that is, Pf small). Therefore. because in most practical
problems we require a small value for Pf, Eq. (38) is a very useful approximation.
22
.1-
1.2 Piro ilmbu lii of DI et•ctio, for .',rling-2 "'rgIl
We now present some numerical -esults for the probability of detecting a
tt Swerling-2 target immersed in Rayleigh noise which varies from pulse-to-pulse
and log-normally distributed clutter which fluctuates from scan-to-scan. By using
Eq. (37), we have calculated the probability of detection Pd 2 for false-alarm prob-L-4 -6
abilities of 10 and 10 . These calculations have been performed, assuming
o ý 0.7, for a number or values of i3 and N, and are presented in Figures 7-16.
For convenience we have also presented in Table 2 the values of the signal-to-noise
ratio c required for Pd2 0. 90, 0. 99 and 0. 999, for a number of different values
of noise-to-clutter ratio i3 and pulses integrated.
The effect on Pd2 of changing the standard deviation a of the log-normal dis-
tribution is illustrated in Figures 17 and 18. We note that a change in a from 0. 7
to 1. 1, results in more than one order of magnitude change in the value of signal
to noise ratio a required for a given probability of detection Pd2" For the case of
vanishing clutter 0.3 oc our numerical data compares well with the data in refer-
ence I for a Swerling-2 target immersed in Rayleigh noise. In the opposite limit
of vanishing noise 13 - 0 we can compare our results with those of Trunk only for
IIN = 1, because Trunk ha-; considered the "a;e when the clutter varies rrom pulse-
to-pulse and we have considered the limit when the clutter varies only from scan-
to--scan.
From Figures 7-18, wz note that large signal-to-noise ratios are usually
required for high detection probabilities. This suggests that it might be useful to
have a large a approximation to (37). For large a, it is shown in Appendix C that
(37) can be approximated as
A cmpaiso ¥°clcuatd (0) om1d2 exp 0 ) ( 0 ' (40)
f• A comparison in Table 3 of numerical results calculated from (40) with the com-
"puted data in Figures 7-18 indicates that (40) is quite an excellent approximation
for Pd2 provided m, fN >> I. This encompa-.ses nearly the entire range of practical
interest (that is, Pd2 > 0. 5). Therefore, in quite a number of practical situations
it is not necessary to numerically evaluate the integrals in (37), and Eq. (40) can
be used.
7. Trunk, G. (1971) Further results on the detection of targets in non-gaussian
sea clutter- IEEE Trans. on Aerospace and i'lectronic Systems AES-7:553.
2:3
!o-
p •o o0 P o 7aO07 to. 0 O 16
10-1 N"i0"o . 0\ ,I l IN N 10W
N 1N O I - N-6 I . 1
'0 1 . , 0 10102 I0$ 104 10• 0 10 t 0o,0
Figure 7. Probability of Detection Figure 8. Probability of4Detection(Swerling 2) for P i0-, 0 3 . 316, (Swerling 2) for Pf - 10 , j3 1.0,a 0. 7 a 0.7
Pf I ' 0 7 P t 10 -4"o" ,' 07 p o " • 07•''3 16 j• ,to
So -,
I0N Io
I .-N 3
10" N 10 10 2--N
NN1
10 I0}o,* 0 N6 10
Figure 9. Probability ofDetection Figure 10. Probability of Detection(Swerling 2) for Pf 10 , = 3.16, (Swerling 2) for P 1b-4, 3 10,Y z 0.7 (= 0.7
24
" , j 4G_00 3O 165
' 00
N 3 O
N 10 -N-6
to-, 7 rN 00.76 o-
10*
Figure 13. P~robability of Detection Figure 14. Probability of Detection(Swerling 2)frP 1-4, ;i 10, (Swerling 2) for P~ 10,1 3
a0. 7 a 0.,0170 6
250
IiIpf~P
Jo-(, 0'0
Pf 1 0 7 / •100 07= I0
10 30
Q•--~-N 6." •
SN'3 10" N
N•IO". N'" O6
10 o
..-..-s.L-.•.•-1
I 10 '0 I Io10 10 , 10
Figure 15. Probability of Detection Figure 16. PI-obability of Detection(Swerling 2) for Pf 10*6, i3 10, (Swerling 2) for P£ 10-6, 13 100,o 0.7 cr 0.7
NN - 11N 1•\ zr--cr .II
N0" N 1 10--
SN'IO__•N '1I0
Pf~P '1'0 '
3 31()0N 10 NNI 0
N 10
S. . . . .I . L ..... ...... 1 l ......J I .IlI
10 '0 0. . , . .. I . .0. I
Figure 17. Effect of o n 1 d2 for Figure 18. Effect of ca on Pd2 for
Pf 10-4, 13 3.16 P f 10-6, 13 3. 16
26
r
Table 2. Value of Signal-to-Noise Ratio a Required for Pd2: 0.9, 0.99 and0. 999 VWhen, 11f 10-Cj andc 0. 7
3 p Pd2 P 0.9 P d2 - 0. 9 9 Pd2 - 0.99
IN
0.316 1. 177 X 104 1.234 x 105 1.239 X 10G
3 4 51.0 3.701 x 103 3.880 X 10 3.900 X 10
""3. 16 1. 185 X 10 1.244 X .04 i.244 X 10O
2 3410.0 4.113 X 10 4.321 X 103 4.341 X 104
100.0 1.329 X 102 1. 403 X 10: 1.410 X14
N :3
40.3 1,(i :3380 8500 1.925 x 10
1.0 1055 2680 6. 100 x 10:
3.16 348 855 1.950 X 10
10.0 113 280 6.40 10 2
100.0 17.7 46 1.07 .102
N G
0.31IF 2350 4150 6650
1.0 640 1300 2120
3. 16 234 412 660
"10.0 75. 5 135 240
100.0 9. 7 17. H 29. 5
N -10
0.3 1 G 1980 3000 43-50
1.0 (;20 937 1:300
.3. 16 197 298 415
10.0 64 96 137
100.0 7. C 12 17
14
27 '
Table 3. Comparison of Pd2 Calculated From Equation (40) With the Exact
Result for a = 0.7 and Pf = 10-6
I P d2
(ad2 (calculated fromi3• N Yo (exact) Equation (40)
1 1000 1 390 0.67 0.68
1 1000 3 1170 0.89 0.89
1 1000 6 2350 0.968 0. 967
1 1000 10 3900 0.994 0.9932
10 100 1 44 0.65 0.65
10 100 3 23 0.875 0.876
10 100 G 240 0.965 0.966
10 100 10 405 0.9922 0.9918
f.3 Prorabilits, of -la.te,ion for y ierling-] oi'argeh e
We now calculate the probability of detecting a Swerling-1 target immersed inlog-normal clutter which varies from scan-t,)---,can and Rayleigh noise which varies
from pulse-to-pulse. By using (36), we have computed the probability of detectionSPd for false-alarm probabilities of 10-4 and 10- 6. These calculations for a - 0. 7
and a number of different values of ;3 and N are presented in Figures 19-28. From
l.igures 19-28, we observe that large signal-to-noise ratios a are usually required
for high probabilities of detection. For large a, it is readily demonstrated that
Pd + -o exp -( N 41)
A comparison of numerical results calculated from (41) with the computed data in
Figures 19-28 indicates that (41) is an excellent approximation for Pdl whenever
Pdll >0.7.
The proof parallels that in Appendix C. In particular, observe that for large othe integrand in (17) can be expanded in a Taylor series about q : 0. This givesIM•) 1 1. Upon setting M(V) 1 in (3G), we can readily do the f. and yintegrations.
28
P, "°0 f 10r0 7 •"07
I. I o .,-8 '0 316 8P "
.o t N 1.3 6 . 6, ,
I ,
N 1.3 6 0 It
SO-• . . . . I .... ... .&..a..a.......aJt • O 1O .010101 10 -L4 i0AIC-3"02 0' - '* 'a'-' 3
0210 0 _ 10 10' 10
V £Figure 19. Probability of Detection Figure 20. Probability of Detection(Swerling 1) for P I0-4, ) 0. 316, (Swerling 1) for Pf 1 i0- 4 , 1. 1,-o= 0.7 = 0.7
I29
a, 07,* io.. 07
;c;;
* o 07
'aN 1'. 3,6. 10 ].oICN'
a N.
i 10
-I FFigure 21. Probability of Detection Figure 22. Probability of Detection(Sw -rl'ng 1) for P 1 0-4~,3 3. 16, (Swerling 1) for Pf 1-4, 3i 10,
a 0.70. 7
- 29
~L
0), 0 O0100
S "D -30103,6,10
10' - '3 0'1
a.;',!l ~ N' 10•'N"6 'NI
iO-2 , iO-Z
11.S , ,,... . ...... 5 . . . . . ................. .. ........... 1 . . ..
10-0 i1 1 i0 10 0' 10 '
Figure 23. Probability of Detection Figure 24. Probability of Detection
(Swerling 1) For P = I0-4, 0 100, (Swerling 1) for P1 = 10-6, 0:0.316,a -0.7 a= 0.7
a 07 Pt ' O' C * 07p '316
10-1 N 1 i. 5,6.10 '0"1
a. a1'
0-2 N 10-
0".................... ...,, .,......I..............0 - ••-
lo.0 0t ' o, -0
Figure 25. Probability of Detection Figure 26. Probability of Detection .--
(Swerling 1) for P1 1 0-6, 1 ' i, (Swerling 1) for Pf = 10-6, .3 = 3. 16,
a:0.7 a 0.7
30
L.
O0 Pf I0*6
-- --
N
S10.0
ao-.7 -= 0.7
it is interesting to note from Figures 19-28 that unless ,l ) 10. Pdl is essen-
S~tially independent of N. This is quite evident from physical reasoning because for
i.• small noise-to-clutter ratios neither the target nor the clutter-plus-noise fluctuate
I • from pulse-to-pulse. Therefore, summing N-pulses cannot offer any, advantage.
i ~~This is also evident from (41) and ('38). F-rom (38) we h]ave tat for Y large
SH( f) N
Sy = -- (42)
i ~where IH is a function of P. If we use this result in (41) we obtain
N 11:-
10- j Y 10
* 1Pdl •exp L--J 0 e --- C-J3 (403)
fa
(Swhich is of N. For l values of ,1, the results in (42) and
t clealy independetoN.Tiisqtevdent larompyiageaoigbeaso
(43) are no longer appropriate because then Y isr no longer large, as is required
Sfor (42) to be valid.
Q
. 31
5. DI-SCI S1_;ON
The results we have presented in Sections 2-4 are strictly valid for a receiver
which consists of a square-law detector and integrator. However, most systems
operating in an environment where clutter is a consideration are likely to employ
an MTI (moving target indicator) to reduce clutter. When an MTI is used our
results are only approximately valid, because the noise* samples are then not
statistically independent from pulse-to-pulse, as we have assumed, (This state-
ment applies to the target-plus-noise for a Swerling-2 target. ) In order to see
this, we consider a single stage MTI. Let nhi_ be the noise present at the input to
the MTI at time (i - O)-r, ni be the noise at time i-, and ni+1 be the input noise at
time ( + O)-r. Then the MTI noise output ni at time i .7 is i. ni- 1 -n and the output
ih at time (i + 1)nr is ni = ni n 1 herefore, even though n- 1 , ni and n
are statistically independent, ir and niý+1 are not, because (i ((ni-I - ni)2 1~ 1-
(ni - ni+l) - (n1) i 0. Consequently, the noise samples at the output of the MTI
are partially correlated from pulse-to-pulse. Thus if NO of these pulses are
integrated, the correct probability of detection will be somewhere between our
answers in Figures 7-28 for N zI (complete correlation of all N0 pulses) and
N = N (complete decorrelation). For a single stage MTI, the pulse-to-pulseo
correlation coefficient is 1/2, for a two stage MTl the correlation coefficient is
2/3, etc. This suggests that the N = I results in Figures 7-28 will be a reasonable
approximation when a multistage MTI is used, even if N-pulses are integrated
following the MTI.
Note that this will not be a problem with the clutter samples because the cluttersamples have been assumed to be completely correlated from pulse-to-pulse,and the MTI does not change this. The clutter statistics from scan-to-scan areof course still log-normal.
**The value of the clutter level which would be used in calculating 13 is the residual
clutter level present at the MTI output terminals,
32
r
B IAVAILAB1E COPY
SL References
1. Alarcum, J. , and Swerling, P. (1960i) Studies of target detection by pulsedradar, IRE Trans. on Inform Theory I' -6:2.
2. Trunk, G., and George, S. (1970) Detection of targets in non-gaussian seaclutter, IEEE Trans. on Aerospace and Electronic Systems AES-:5620.
: 3. Schleher, D. (1975) Radar detection in Iog-no)rnial clutter-, IEEE InternationalRadar Conference, IEEE Press (Publication No. 759 ('140938-1 AE2S), Ne%%
. ~4. B~eckmann, P. (1967) Probability in Communication Engineering, Harcourt,
SBrace and World, Nem, York.
5. Erdelyi, A., et al (1954) Tabbles of Integral Transform (Bateman U~anuscripItAPrjet0 McGraw-Hill, New York.
:• 6. B•erkowitz, IR. (1965) Modern ]Radar, Wiley, New York.
'7. Trunk, G. (1971) Purther results on the detection of" targets in non-gaussiansea clutter, IEEE Trans. on Aerospace and Electronic Sy¢stems AES-7:553.
33
BET2AVAILABLE Copy
Appendix AA
Pr~obaility Density for a Constant Target
Here we give the probability density for the envelope squared of the sum ofN-independent siamples rrorn a constant target immersed in Rayleigh noise %%hichvaries from pulse-to-pulse and log-normal clutter which varies fr'om scan-to-scan.
For Y :! 0, the result is
P()(27)3/ f dF e-ýGN(,Y0o
(A 1)
dii exp In 2 ji(2ý -2 3cr2 a +1' a2 N 1
2g- 2-',' 2 ag 1 2 + a 2
1"'2where a N ~C/Gny C is the (constant) target voltage and 3,Y, GN etc. are asdefined previously.
35Sa
.-- -- '-,.
Appendix B
Derivation of Equation (38)
If we let n Y/Y0 and E Yot, we can rewrite (35) as
Y P ( -)'/2 fN-I [2Yo(rIt)2 1 '2
2f ( ,'2 f T- aN-1
(Bi1)exp -Yoi1+ t)• 2 (n 2Y. t)}
For Y >> 1, we can use the asymptotic expansion for IN- (Z). If this is done and
we also make the substitution t = x2 and r v2 , we can rewrite (Dl) as
P (2Y / o V 1/2 dv f dx (v )1 e -Y (x-V)2 1 n2
-fr 2;3 /2 exp 0 -v
(B2)
For Yo co it is clear that the integrand of the x integral is sharply peaked about
x = ov, because of the term exp [-Y(x - v) 2. Therefore, we can set x =v in the0 2
entire x integrand, except for exp [-Y (x - v) to get
37
o 2 12 d J exp 1, 213 0 1 - ze-Yoz2 (3
v 2.
where in the last integral in (B3) we have replaced x - v by z and extended the
range of integration on z to the entire real line. If we perform the z integral and
then let u (2)-1 2 a In [v(2Y 01N)1/21 we get
1 0 f -2 du (B34)
Pf 1. 2 efB). -
where p (B a 2 ) -1/2 n 2 N). Equation (38) fullows immediately from (1M4).0
j 3
.I.
38,
Appendix C
Derivation of Equotion (40)
For large a, it is convenient to rewrite (37) as
Y
0 0
~d2 2a2)' f e- Y J ~f T N-1 [2(Y) 1 J/o
(CI1)
exp 1 ( , 2(
L 8a \ N J
As a o , it is clear that momt of the contribution to the integral Comeýi from -
near zero. Therefore, we may expand IN 1(...) in a Taylor s.eries about • 0 to
obtain
Y___ 1
1 . (1 1 o d - Y 1' N - d e x p - 1o i n'n (2 -L3d2 1 (2r 1)/2 f (N- ! f 8a 2 N302
0a
(C.2)
39l 3 9 3
' *i . . . . . .-- - - .,
If we recall that
I exp [ In 2(Aý 12(2r) /2a
we see that (C2) becomes
y0
-p 1 I - a dY e-Y yN.-1-d2 - (N-)' (C3)
Finally, if we use form 5 on p. 134 of Ref. 1, we can evaluate the integral in (C3)
to yield (40).
:1-
1. Erdelyi, A., et al (1954) Tables of Integral Transform (Bateman ManuscriptProject) McGraw-Hill, New York,
40
S*-~.. ... ~ .. ,.. .~..'~Jt'XI.A~t S .fl~flS 'fl. .~ n~slJ~ a___ .
WIT1IC SYITEIB
BASS UNrrSsQuantity UniiSjyqpbol orml
lon'lth metre mum IIal% kilgrmk
time second aelectric current ampere Athermodynamic temperature kelvin Kamount of subetance mole MCIluminous intensity cendoea *d
SU.PULAMJTAY UNIT;plane angle radian redsolid angile eteradian at
DIVU) UNIT;Acceleration motre per weond equared .. MISactivity (of a radioactive eourcel dilsitepstion par second Idisintepratioullaangular acceleration radian per second squated aft/angular velocity radian per second radieama Square metre .. e ui er gMdensity kilogpt uicMar mfelectric capacitance feTad F A-elVelcvtrical conductanre ielement S ANeectric field strength volt per metre .. Vim
electric inducs.nces henury H V-slAelectric potential diffaenmce volt V WIAelectric reeistance ohm VIAelectromotive force volt V WIAenergy joul~e I N-mentropy joule per kelvin ..
force newton N kt.nr/a1frequency hertz HZ (cycleysIlluminirance lux lx kuhnluminnc candela per square metr .. dhnlumino0us flux: lumen Im a"~emagnetic field strength ampere per metars,. A/nemagnetic flux weber Wi. V-3magnetic flux density tfisia T Whimmagnetomotl'te force ampere Apower watt W YS
pressre pa Ia I's WN,quantity of electricity coulomb C A-squantity of heat joule IN-mradiant intensity welt per ateredien .. Wiltapecilic beat )oute par kilogram-kelvin .. hkg"Ksetres pascal Pa Win,tharmnal conductivity welt pet metro-kelvin .. Wlm.Kvelocity metre pet weond ... isvviscosity. dynamic pascal-waond Feeviscosity, kinematic square metr per second mulsvoltage volt V WIAvolume cubic matte Mwavenumbe reciprocal meetre j:sv@Ymu
.1, joule
3i PRFIXES:
_Autltlication Factors Pranx SI Yrnho
I ON0 00 0 00 0 0 0 1 0 :1ar T1000000000-10 gipgs
1000 000 10. Meap M1000 103 kilo I
100- 100 hectnl h10-0 doka* do
0.1 -10-' daci d0.01 - 10-1 mail, c
0.001 0- 10 milli In0.000 001 -10'4 microA
0.000 000001 *10-9 neow n0. 000000 000001 0-1Ic
0.0W000 000 00000001-10-"1 I0.000 000 000 000 000 001- 10- 1 altoa
To be avoided where possible.
